mathlib3 documentation

ring_theory.kaehler

The module of kaehler differentials #

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Main results #

Future project #

@[reducible]
def kaehler_differential.ideal (R : Type u_1) (S : Type u_2) [comm_ring R] [comm_ring S] [algebra R S] :

The kernel of the multiplication map S ⊗[R] S →ₐ[R] S.

def derivation.tensor_product_to {R : Type u_1} {S : Type u_2} [comm_ring R] [comm_ring S] [algebra R S] {M : Type u_3} [add_comm_group M] [module R M] [module S M] [is_scalar_tower R S M] (D : derivation R S M) :

For a R-derivation S → M, this is the map S ⊗[R] S →ₗ[S] M sending s ⊗ₜ t ↦ s • D t.

Equations
theorem derivation.tensor_product_to_tmul {R : Type u_1} {S : Type u_2} [comm_ring R] [comm_ring S] [algebra R S] {M : Type u_3} [add_comm_group M] [module R M] [module S M] [is_scalar_tower R S M] (D : derivation R S M) (s t : S) :

The kernel of S ⊗[R] S →ₐ[R] S is generated by 1 ⊗ s - s ⊗ 1 as a S-module.

def kaehler_differential (R : Type u_1) (S : Type u_2) [comm_ring R] [comm_ring S] [algebra R S] :
Type u_2

The module of Kähler differentials (Kahler differentials, Kaehler differentials). This is implemented as I / I ^ 2 with I the kernel of the multiplication map S ⊗[R] S →ₐ[R] S. To view elements as a linear combination of the form s • D s', use kaehler_differential.tensor_product_to_surjective and derivation.tensor_product_to_tmul.

We also provide the notation Ω[S⁄R] for kaehler_differential R S. Note that the slash is \textfractionsolidus.

Equations
Instances for kaehler_differential
@[protected, instance]
def kaehler_differential.module (R : Type u_1) (S : Type u_2) [comm_ring R] [comm_ring S] [algebra R S] :
@[protected, instance]
@[protected, instance]
def kaehler_differential.nonempty (R : Type u_1) (S : Type u_2) [comm_ring R] [comm_ring S] [algebra R S] :
@[protected, instance]
def kaehler_differential.module' (R : Type u_1) (S : Type u_2) [comm_ring R] [comm_ring S] [algebra R S] {R' : Type u_3} [comm_ring R'] [algebra R' S] [smul_comm_class R R' S] :
Equations
@[protected, instance]
@[protected, instance]
def kaehler_differential.is_scalar_tower_of_tower (R : Type u_1) (S : Type u_2) [comm_ring R] [comm_ring S] [algebra R S] {R₁ : Type u_3} {R₂ : Type u_4} [comm_ring R₁] [comm_ring R₂] [algebra R₁ S] [algebra R₂ S] [has_smul R₁ R₂] [smul_comm_class R R₁ S] [smul_comm_class R R₂ S] [is_scalar_tower R₁ R₂ S] :
@[protected, instance]

The quotient map I → Ω[S⁄R] with I being the kernel of S ⊗[R] S → S.

Equations
def kaehler_differential.D (R : Type u_1) (S : Type u_2) [comm_ring R] [comm_ring S] [algebra R S] :

The universal derivation into Ω[S⁄R].

Equations
def derivation.lift_kaehler_differential {R : Type u_1} {S : Type u_2} [comm_ring R] [comm_ring S] [algebra R S] {M : Type u_3} [add_comm_group M] [module R M] [module S M] [is_scalar_tower R S M] (D : derivation R S M) :

The linear map from Ω[S⁄R], associated with a derivation.

Equations
@[simp]
theorem derivation.lift_kaehler_differential_comp_D {R : Type u_1} {S : Type u_2} [comm_ring R] [comm_ring S] [algebra R S] {M : Type u_3} [add_comm_group M] [module R M] [module S M] [is_scalar_tower R S M] (D' : derivation R S M) (x : S) :
@[ext]
theorem derivation.lift_kaehler_differential_unique {R : Type u_1} {S : Type u_2} [comm_ring R] [comm_ring S] [algebra R S] {M : Type u_3} [add_comm_group M] [module R M] [module S M] [is_scalar_tower R S M] (f f' : Ω[SR] →ₗ[S] M) (hf : (f.comp_der) (kaehler_differential.D R S) = (f'.comp_der) (kaehler_differential.D R S)) :
f = f'

The S-linear maps from Ω[S⁄R] to M are (S-linearly) equivalent to R-derivations from S to M.

Equations

The endomorphisms of Ω[S⁄R] corresponds to sections of the surjection S ⊗[R] S ⧸ J ^ 2 →ₐ[R] S, with J being the kernel of the multiplication map S ⊗[R] S →ₐ[R] S.

Equations
noncomputable def kaehler_differential.ker_total (R : Type u_1) (S : Type u_2) [comm_ring R] [comm_ring S] [algebra R S] :

The S-submodule of S →₀ S (the direct sum of copies of S indexed by S) generated by the relations:

  1. dx + dy = d(x + y)
  2. x dy + y dx = d(x * y)
  3. dr = 0 for r ∈ R where db is the unit in the copy of S with index b.

This is the kernel of the surjection finsupp.total S Ω[S⁄R] S (kaehler_differential.D R S). See kaehler_differential.ker_total_eq and kaehler_differential.total_surjective.

Equations

The (universal) derivation into (S →₀ S) ⧸ kaehler_differential.ker_total R S.

Equations
def derivation.comp_algebra_map {R : Type u_1} [comm_ring R] {M : Type u_3} [add_comm_group M] [module R M] (A : Type u_4) {B : Type u_5} [comm_ring A] [comm_ring B] [algebra R A] [algebra R B] [algebra A B] [is_scalar_tower R A B] [module A M] [module B M] [is_scalar_tower A B M] (d : derivation R B M) :

For a tower R → A → B and an R-derivation B → M, we may compose with A → B to obtain an R-derivation A → M.

Equations
def kaehler_differential.map (R : Type u_1) (S : Type u_2) [comm_ring R] [comm_ring S] [algebra R S] (A : Type u_4) (B : Type u_5) [comm_ring A] [comm_ring B] [algebra R A] [algebra R B] [algebra A B] [algebra S B] [is_scalar_tower R A B] [is_scalar_tower R S B] [smul_comm_class S A B] :

The map Ω[A⁄R] →ₗ[A] Ω[B⁄R] given a square A --→ B ↑ ↑ | | R --→ S

Equations
theorem kaehler_differential.map_D (R : Type u_1) (S : Type u_2) [comm_ring R] [comm_ring S] [algebra R S] (A : Type u_4) (B : Type u_5) [comm_ring A] [comm_ring B] [algebra R A] [algebra R B] [algebra A B] [algebra S B] [is_scalar_tower R A B] [is_scalar_tower R S B] [smul_comm_class S A B] (x : A) :
noncomputable def kaehler_differential.map_base_change (R : Type u_1) [comm_ring R] (A : Type u_4) (B : Type u_5) [comm_ring A] [comm_ring B] [algebra R A] [algebra R B] [algebra A B] [is_scalar_tower R A B] :

The lift of the map Ω[A⁄R] →ₗ[A] Ω[B⁄R] to the base change along A → B. This is the first map in the exact sequence B ⊗[A] Ω[A⁄R] → Ω[B⁄R] → Ω[B⁄A] → 0.

Equations
@[simp]
theorem kaehler_differential.map_base_change_tmul (R : Type u_1) [comm_ring R] (A : Type u_4) (B : Type u_5) [comm_ring A] [comm_ring B] [algebra R A] [algebra R B] [algebra A B] [is_scalar_tower R A B] (x : B) (y : Ω[AR]) :